I read that book recently and I enjoyed it a lot, it is a very well writen book about Bob Hope.
I wouldn't focus on few points that are phrased as opinions. It is the author's perogative to distinguish facts and opinions and Zoglin gets A+ here.

The book is also well sourced, a biographer cannot be responsible fully for all the information. He says that this was said by this particular source, I agree with you that the source might have lied, misremembered and perhaps misquote the phrase.

I also notice a few minor details that were not accurate, indeed it is factual correct that Bing according to his butler had invited Bob and Dolores Hope for dinner once, and he had even stay in Hope's house when his Toluca Lake residence was burned. 
If I am remembering it right, Bing Crosby included some country club stories with Bob, suggesting they were socializing a bit more than Zoglin suggests.

I believe that Bing and Bob had a complicated relationship. They were in my view friendly, as it is supported by 

Here it is also important that the subject is Bob Hope, the Bing Crosby mentions are contextual so I would judge them with the same weight if they were coming from a Crosby biographer.

Bing declined a cameo role as Father O’Malley (it was Hope’s idea), he was less willing to participate in some projects.
So, it seems likely that he might have some problems or complaints from the long standing relationship and their different personalities but not to the extent of really not liking him. 

Bing at some point offered Bob the chance to purchase neighboring vacation houses. 

So even if Bob had some complaints that Bing was perhaps less commited to him cannot reach the point of really not liking him.

Bob also used more Crosby references in his work, he used to do more Crosby jokes while Bing did that few and far in between times.

Bob also used Bing many times as cameo while Bing didn't at the same degree.

Richard Zoglin is also mentioning in his writings that Bing and Bob really came closer when they stayed with families together in UK during the last road picture.

I would say that this is a full life story coverage of Bob and a pleasure to read.